# Lebesgue integral of a_p

1. Mar 24, 2006

### eljose

Let define the function:

$$a_{p}(x)= 1$$ if x is an integer and prime and 0 elsewhere, my

question is.........what would be its Lebesgue integral let,s say from [c,d] with c and d positive and real..

2. Mar 24, 2006

### HallsofIvy

Staff Emeritus
Since this function is only non-zero on only a finite number of points between c and d, isn't the integral obviously 0?

3. Mar 24, 2006

### matt grime

It's Riemann (and hence lebesgue) integral is rather trivially zero on any interval. My question is why would you need to ask this?

4. Mar 25, 2006

### eljose

then why the integral of the function $$f(x)=1$$ iff x is rational and 0 elsewhere is different from 0?....

5. Mar 25, 2006

### shmoe

You're sayig the Lebesgue integral of this function is non zero? How do you figure?

6. Mar 25, 2006

### matt grime

1 on a set of measure zero 0 every where else, aka almost everywhere zero. that the integral is zero of such a thing is practically the point of lebesgue theory.

7. Mar 25, 2006

### HallsofIvy

Staff Emeritus
It isn't! Who told you that it was? The lebesque integral of the function you give is 0 over any finite interval.

IF, instead, you define f(x)= 1 if x is irrational and 0 if x is rational (1- your f(x)) then the integral of f over the interval [a, b] is b-a.

8. Apr 2, 2006

### eljose

i know i have posted this topic or analogue before but i have the doubts with lebesgue integration:

a) the Lebesgue integral of exp(x)..is equal to Riemann integral of exp(x)

b) $$D_{t}\int_{0}^{t}d\mu{f}= f ?$$

c)what would be the formula for integration by parts in Lebesgue integration?..

thanks.

9. Apr 2, 2006

### matt grime

integration by parts requires the integrands to be differentiable or to be a derivative and hence continuous, so there is no point in using lebesgue integration, is there?